If two graphs are isomorphic then will their determinants be equal?

It is indeed true that the adjacency matrix of isomorphic graphs will have the same determinant. Indeed, if $A = PBP^T$ for a permutation matrix $P$, then $$ \det(A) = \det(P) \det(B) \det(P^T) = \det(P)^2 \det(B) = \det(B). $$ In fact, we can say more: because $A$ and $B$ are similar matrices, they will have the same eigenvalues. Note that if two matrices have the same eigenvalues, then they automatically have the same determinant.

The converse does not hold: if $A,B$ are the adjacency matrices of two graphs and we have $\det(A) = \det(B)$, it does not necessarily hold that the graphs are isomorphic. In fact, even if $A$ and $B$ have the same eigenvalues, it does not necessarily hold that the graphs are isomorphic. Examples of this phenomenon are called isospectral (or cospectral) pairs of graphs.