Cue-induced drug craving progressively intensifies after withdrawal from self-administration of cocaine, methamphetamine, and other drugs of abuse, a phenomenon termed incubation of craving. For cocaine and methamphetamine, expression of incubated craving ultimately depends on strengthening of nucleus accumbens (NAc) synapses through an accumulation of high conductance Ca2+-permeable AMPA receptors (CP-AMPARs) that is detectable with electrophysiological approaches. This study sought to further characterize glutamate receptor adaptations in NAc core during methamphetamine incubation. Previous biochemical studies revealed that the CP-AMPARs accumulating after cocaine incubation are mainly homomeric GluA1 receptors and that their accumulation is reflected by increased cell surface GluA1. Here, for methamphetamine, we observed no significant change in surface or total GluA1 (GluA2 and GluA3 were also unchanged). Nonetheless, GluA1 translation was elevated after incubation of methamphetamine craving, as recently found for cocaine. Additionally, for cocaine, we previously observed a withdrawal-dependent decrease in mGlu1 surface expression that precedes and enables CP-AMPAR accumulation and incubation of craving, reflecting weakening of mGlu1-dependent mechanisms that normally limit synaptic CP-AMPAR levels in the NAc core. Here, we observed no change in surface or total mGlu1 protein or its coupling to Homer scaffolding proteins after methamphetamine withdrawal, nor did elevation of mGlu1 tone through repeated injections of an mGlu1-positive allosteric modulator delay incubation of craving. These findings suggest a common role for increased GluA1 translation, but not decreased mGlu1 function, in the incubation of methamphetamine and cocaine craving. We speculate that increased GluA1 translation near synapses may drive formation and synaptic insertion of homomeric GluA1 receptors in the absence of detectable changes in GluA1 protein levels.