The generic hypersurface of degree 5 in P^3 has no rational curves at all, let alone a line or a conic:  see

G. Xu, "Subvarieties of general hypersurfaces in projective space", J. Diff Geom 1994

So you can certainly get a complex such hypersurface without any lines or conics just by embedding the function field of the moduli space of hypersurfaces into C.  Since the locus of quintics containing a line or conic is a proper closed subvariety, you'll also get plenty of examples over finite fields if you make the finite field large enough (though to be fair this will not exactly be "explicit.")

I guess the generic quartic hypersurface probably also has no lines or conics (though it will have rational curves) but I didn't check this.