This is true if the anticanonical system (which you call the fundamental system) is *very ample*. Indeed one can take a Lefschetz pencil and apply Theorem 1.4 of SGA 7, Exposé XIX (Noether's theorem by Deligne). Now we know the complete list of Fano threefolds with Picard number 1 , thanks to Iskovskikh; there are only 2 or 3 cases where $-K$ is not very ample, maybe they can be checked by hand.

Afterthought : Actually the result does *not* hold for at least one family of Fano threefolds with Picard number 1, namely those for which the anti canonical map $\pi:X\rightarrow Q\subset {\Bbb P}^4$ is a double covering of a smooth quadric in ${\Bbb P}^4$. Indeed the smooth surfaces in $\ |-K_X|\ $ are of the form $S_h:=\pi^{-1}(Q_h)$, where $Q_h$ is a smooth hyperplane section of $Q$. Since $\mathrm {Pic}(Q_h)=\Bbb{Z}^2$, the Picard number of $S_h$ is $\geq 2$.