Jelly, 7 bytes

ZẆ$⁺€Ẏċ

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How it works

ZẆ$⁺€Ẏċ  Main link. Arguments: B, A



  $      Combine the two links to the left into a monadic chain.

Z          Zip; transpose the matrix.

 Ẇ         Window; yield all contiguous subarrays of rows.

   ⁺     Duplicate the previous link chain.

    €    Map it over the result of applying it to B.

         This generates all contiguous submatrices of B, grouped by the selected

         columns of B.

     Ẏ   Tighten; dump all generated submatrices in a single array.

      ċ  Count the occurrences of A.

Brachylog (v2), 10 bytes

{{ss}ᵈ}ᶜ

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I like how clear and straightforward this program is in Brachylog; unfortunately, it's not that short byte-wise because the metapredicate syntax takes up three bytes and has to be used twice in this program.

Explanation

{{ss}ᵈ}ᶜ

  s         Contiguous subset of rows

   s      Contiguous subset of columns (i.e. transpose, subset rows, transpose)

 {    }ᵈ    The operation above transforms the first input to the second input

{       }ᶜ  Count the number of ways in which this is possible

MATL, 12 bytes

ZyYC2MX:=XAs

Inputs are A, then B.

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Explanation

Consider inputs [1,4; 3 1], [3,1,4,5; 6,3,1,4; 5,6,3,1]. The stack is shown with the most recent element below.

Zy    % Implicit input: A. Push size as a vector of two numbers

      % STACK: [2 2]

YC    % Implicit input: B. Arrange sliding blocks of specified size as columns,

      % in column-major order

      % STACK: [3 6 1 3 4 1;

                6 5 3 6 1 3;

                1 3 4 1 5 4;

                3 6 1 3 4 1]

2M    % Push input to second to last function again; that is, A

      % STACK: [3 6 1 3 4 1;

                6 5 3 6 1 3;

                1 3 4 1 5 4;

                3 6 1 3 4 1],

               [1 4;

                3 1]                    

X:    % Linearize to a column vector, in column-major order

      % STACK: [3 6 1 3 4 1;

                6 5 3 6 1 3;

                1 3 4 1 5 4;

                3 6 1 3 4 1],

               [1;

                3;

                4;

                1]  

=     % Test for equality, element-wise with broadcast

      % STACK: [0 0 1 0 0 1

                0 0 1 0 0 1;

                0 0 1 0 0 1;

                0 0 1 0 0 1]

XA    % True for columns containing all true values

      % STACK: [0 0 1 0 0 1]

s     % Sum. Implicit display

      % STACK: 2

05AB1E, 10 bytes

øŒεøŒI.¢}O

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øŒεøŒI.¢}O     Full program. Takes 2 matrices as input. First B, then A.

øŒ             For each column of B, take all its sublists.

  ε     }      And map a function through all those lists of sublists.

   øŒ          Transpose the list and again generate all its sublists.

               This essentially computes all sub-matrices of B.

     I.¢       In the current collection of sub-matrices, count the occurrences of A.

         O     At the end of the loop sum the results.

Dyalog APL, 6 4 bytes

≢∘⍸⍷

This is nearly a builtin (thanks H.PWiz and ngn).

  ⍷       Binary matrix containing locations of left argument in right argument

≢∘⍸       Size of the array of indices of 1s

Alternative non-builtin:

{+/,((*⍺)≡⊢)⌺(⍴⍺)*⍵}

Dyadic function that takes the big array on right and subarray on left.

                  *⍵       exp(⍵), to make ⍵ positive.

    ((*⍺)≡⊢)⌺(⍴⍺)        Stencil;

                            all subarrays of ⍵ (plus some partial subarrays

                            containing 0, which we can ignore)

               ⍴⍺             of same shape as ⍺

     (*⍺)≡⊢                   processed by checking whether they're equal to exp(⍺).

                           Result is a matrix of 0/1.

   ,                     Flatten

 +/                      Sum.

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Charcoal, 36 bytes

ＩΣ⭆⊕⁻ＬηＬθ⭆⊕⁻Ｌ§η⁰Ｌ§θ⁰⌊⭆θ⭆ν⁼π§§θ⁺κξ⁺μρ

Try it online! Probably should be much shorter than this but Equals isn't working for arrays at the moment. Explanation:

⭆⊕⁻ＬηＬθ

Calculate and loop over the range of allowed values for the top coordinate of submatrices of B that are the same size as A.

⭆⊕⁻Ｌ§η⁰Ｌ§θ⁰

Repeat for the left coordinate.

⌊⭆θ⭆ν⁼π§§θ⁺κξ⁺μρ

Compare each element of A with the appropriate element from the submatrix of B and take the minimum. (As the comparison results are either 0 or 1 it's safe to use StringMap here to flatten the result and get the minimum in a single byte.) This will be 1 for a match and 0 for a mismatch.

ＩΣ

As I've used StringMap to loop over the submatrices I can simply take the digital sum, thereby counting the matches, and cast that to string for output.

Clean, 118 97 95 bytes

import StdEnv,Data.List

?x=[transpose y\z<-tails x,y<-inits z]

$a b=sum[1\x<- ?b,y<- ?x|y==a]

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Python 2, 101 bytes

lambda a,b:sum(a==[l[j:j+len(a[0])]for l in b[i:i+len(a)]]for i,L in e(b)for j,_ in e(L))

e=enumerate

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JavaScript (ES6), 93 bytes

Takes input as (A)(B).

a=>b=>b.map((r,y)=>r.map((_,x)=>s+=!a.some((R,Y)=>R.some((v,X)=>v!=(b[y+Y]||0)[x+X]))),s=0)|s

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R, 95 bytes

function(A,B,x=dim(A),D=dim(B)-x){for(i in 0:D)for(j in 0:D[2])F=F+all(B[1:x+i,1:x[2]+j]==A);F}

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Python 2, 211 bytes

a,b=input()

l,w,L,W,c=len(a),len(a[0]),len(b),len(b[0]),0

for i in range(L):

 for j in range(W):

  if j<=W-w and i<=L-l:

   if not sum([a[x][y]!=b[i+x][j+y]for x in range(l)for y in range(w)]):

    c+=1

print c 

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Fairly straightforward. Step through the larger matrix, and check if the smaller matrix can fit.

The only even slightly tricky step is the list comprehension in the 6th line, which relies on Python's conventions for mixing Boolean and integer arithmetic.

Groovy, 109 bytes

{a,b->(0..<b.size()).sum{i->(0..<b[i].size()).count{j->k=i-1

a.every{l=j;k++

it.every{(b[k]?:b)[l++]==it}}}}}

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Scala, 151 bytes

(a,b)=>{(0 to b.size-a.size).map(i=>(0 to b(0).size-a(0).size).count(j=>{var k=i-1

a.forall(c=>{var l=j-1;k+=1

c.forall(d=>{l+=1

b(k)(l)==d})})})).sum}

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