I would say not really. SSDs will help with app initialization and loading of local assets, but when it comes large and complex models, you are more likely CPU-bound as Fusion 360 unpacks the model and begins associating joints, bodies, and possibly design history.
That said, SSDs are easily one of the best improvements to personal computing in the last decade - if you haven't gotten an SSD yet, I highly recommend it at least for your boot drive. Anecdotally, I've played with whether to install certain apps on my SSDs vs HDDs at home, and there really just isn't enough disk IO during regular Fusion 360 operations to stress a platter drive, unless maybe it's a 5400rpm drive with a tiny cache.
The biggest impact when adding an SSD is going to be your boot drive where Windows [or macOS] are installed, because everything has to talk to Windows and utilize its DLLs at some point. Given the current state of tech and file sizes, I'd say 256GB is the bare minimum for a boot drive, but 512GB or greater would be better. For a storage SSD, a 2TB budget model is not that spendy these days.
But, again, that's really just going to help Fusion 360 initialize. Your CPU, GPU, and RAM generally have a stronger impact on performance.
Finally, note that an SSD is probably going to be slightly speed-limited by USB 3.0 transfer rates. Thunderbolt, PCIe, and USB 3.1+ connections have more than enough headroom.
Hope this helps!
Lance Carocci
Fusion QA for UI Framework/Cloud Workflows, and fervent cat enthusiast