I’d push back on this one, shifting the goal to be getting the engineering teams to all adopt a quality mindset.
Go in with an open mind and assess where the company is at - how are they working now, what is working for them, and what isn’t working for them. I’ve seen too many people go in to this kind of task that think that everyone needs to do a certain JIRA workflow, that bug reports have to be done in some form or another, etc, etc, but improving quality is the same as any other problem you can use agile to solve. You communicate and you iterate, making incremental improvements to build towards big changes.
I disagree with the idea of needing management/leadership’s buy in to dictate change. Sure, that can be useful, but it’s much better if you can build the support organically - i.e. if you think the biggest issue is that bugs aren’t tracked well, spinning up a bug tracking system with lots of required fields and things is setting yourself up for failure. Focus on true MVP, i.e. just use a very base install of a bug tracker (or maybe even a spreadsheet) to start, with minimal required fields - maybe just title/description/severity, and iterate and go from there.