The magnetic stability of E′ centers and the peroxy radical on the surface of α-quartz is investigated with first-principles calculations to understand their role in magnetic flux noise in superconducting qubits (SQs) and superconducting quantum interference devices (SQUIDs) fabricated on amorphous silica substrates. Paramagnetic E′ centers are common in both stoichiometric and oxygen deficient silica and quartz, and we calculate that they are more common on the surface than the bulk. However, we find the surface defects are magnetically stable in their paramagnetic ground state and thus will not contribute to 1/f noise through fluctuation at millikelvin temperatures.