We study surface critical phenomena of the
S4 model having Gaussian randomness. Using the renormalization-group 4-d expansion, we obtain surface critical exponents indicating the decay of the correlation function, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ and ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$, at the random fixed point. These exponents are evaluated for both the ordinary and special transitions up to the second order, and the surface scaling relation ${\mathrm{\ensuremath{\eta}}}^{\mathrm{random}}$=2${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\perp}}}^{\mathrm{random}}$-${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\parallel}}}^{\mathrm{random}}$ is confirmed in this random system. We also confirm the conformal invariance of the real-space correlation function at criticality.