I am going to use XXYY as the number to avoid subscripts.
XXYY is a multiple of N+1, XXYY = (N+1)*(X0Y). Because XXYY is a square it must be a multiple of (N+1)^2. This imples X+Y=N+1
Also, (N+1)*(AB) = X0Y, which implies AB0+AB=X0Y. Then A=X-1 and B=Y. If B=4 and A=N-4 then AB=N*(N-4)+4 = (N-2)^2. This is a perfect square. Therefore for any base N>=5, (X,Y)=(N-3,4) will produce a desired perfect square whose value will equal [(N-2)*(N+1)]^2.
But these are not the only squares. Substituting expressions for X and Y into perfect square AB yields (X-1)^2+(X-1)*Y+Y. The smallest nontrivial perfect square occurs when X=3 and Y=7, implying 3377 base 9 is a perfect square, 3377 base 9 = 2500 = 50^2.
Further nontrivial solutions include (using A=10, B=11, etc.) 44AA base 11, 8899 base 16, 33FF base 17, 55DD base 17, 44II base 21, 55GG base 21, etc.