With the AstraZeneca vaccine (and presumably J&J would be similar) a consideration is that the side effects you see might actually have to do with the platform itself.
Unlike the mRNA platforms, these vaccines use an existing virus to make the spike proteins. Specifically, an Adenovirus (so the sort of thing that might be implicated in some colds) but which ordinarily thrives in Chimpanzees instead of Humans, so they can have confidence you aren't immune to it already. The virus is altered to tell human cells to make the protein chosen by the engineers (for COVID-19 this was the spike protein from the SARS-CoV-2 virus); and not more copies of itself, once the copies in the vaccine injection are cleaned up (typically a few hours to a few days) the virus is gone, so you can't "catch" the virus.
But of course your immune system has no particular reason not to destroy this (harmless) Chimp virus meanwhile, it shouldn't be in your body, so it's every bit as much a target as the spike proteins we want immunity against.
Thus, it's certainly possible any fever one particular patient suffers is a side effect from their immune defence against this poor harmless chimp adenovirus, and nothing to do with SARS-nCOV-2 or its spike protein. No way to tell for sure I believe.