The Associated Laguerre Polynomial orthonormality relation reads:
\begin{align} \int_0^{\infty} dx f^k_n(x) f^k_m(x) = \delta_{n,m} \end{align} Where, $$ f^k_n(x) = \sqrt{\frac{n!}{(n+k)!}} e^{-x/2} x^{k/2} L_n^k(x) $$
Is there a way to directly calculating $f^k_n(x)$ without evaluating $L_n^k(x)$ first?
You may take advantage of their generating function. Indeed, the Laguerre polynomials are gathered together as the "coefficients" of the series
$$ F_x(t) := \sum_{n=0}^\infty L_n^k(x)t^n = \frac{e^{-xt/(1-t)}}{(1-t)^{k+1}}, $$ which can be then recovered thanks a contour integral, namely $$ L_n^k(x) = \oint_{C_r} \frac{F_x(t)}{t^{n+1}} \frac{\mathrm{d}t}{2\pi i}, $$ where $C_r$ is a circle centered at the origin with radius $r < 1$, hence $$ f_n^k(x) = \sqrt{\frac{n!}{(n+k)!}}\, x^{k/2}e^{-k/2} \oint_{C_r} \frac{F_x(t)}{t^{n+1}} \frac{\mathrm{d}t}{2\pi i}, $$ which you can evaluate thanks to the residue theorem.