Multiplying between generators degree I have to elements of multiplicative group of finite field with generator g - 
$$g^x,g^y $$
Can I get? $$g^{xy} $$
The problem 'given a group and the elements $g, g^x, g^y$, find $g^{xy}$' is known as the Diffie-Hellman problem (or, more precisely, the computational Diffie-Hellman problem).

As for how difficult it is, well, we typically work with groups where this is difficult (by design; we often need to assume that the DH problem is hard).

As for how it might be solved, well, in a finite field multiplicative group, the best way of solving it is to solve the discrete log problem; this is, given $g, g^x$, recover $x$ (and from there, you can compute $(g^y)^x = g^{xy}$.  If you are in a group where recovering $x$ (or $y$) directly is infeasible, then solving the DH problem (to the best of our knowledge) is also infeasible.