Can cognitive impairments and aphasia be symptoms of migraines or is it another issue to look into?

Depends on timing. Confusion and difficulty speaking can be part of a migraine aura, and are often a component of a complicated migraine. If these symptoms persist between migraines, however, they may represent another process.
If new, recheck. Severe or complicated migraines can sometimes cause temporary cognitive impairments or aphasia, but this is pretty rare. Other causes (tia, stroke, anxiety) may be more likely. If these are new, worsening or changing symptoms, or if you are not satisfied with your current treatment plan, you should see a neurologist for further detailed evaluation.