The structural, electronic, and adhesive properties of Cu/SiO$_2$ interfaces are investigated using first-principles density-functional theory within the local density approximation. Interfaces between fcc Cu and $\alpha$-cristobalite(001) surfaces with different surface stoichiometries are considered. Interfacial properties are found to be sensitive to the choice of the termination, and the oxygen density at the substrate surface is the most important factor influencing the strength of adhesion. For oxygen-rich interfaces, the O atoms at the interface substantially rearrange after the deposition of Cu layers, suggesting the formation of Cu-O bonds. Significant hybridization between Cu$-d$ and O$-p$ states is evident in site-projected density of states at the interface. As oxygen is systematically removed from the interface, less rearrangement is observed, implying weaker adhesion. Computed adhesion energies for each of the interfaces are found to reflect these observed structural and bonding trends, leading to the largest adhesion energy in the oxygen rich cases. The adhesion energy is also calculated between Cu and SiO$_2$ substrates terminated with hydroxyl groups, and adhesion of Cu to these substrates is found to be considerably reduced. This work supports the notion that Cu films can adhere well to hydroxyl-free SiO$_2$ substrates should oxygen be present in sufficient amounts at the interface.