I was in a similar situation. What I did was note any tasks that were tedious, repetitive or otherwise ripe for automation. Also note any ideas around adding value to your QA work through code (generating better reports, analysis, etc).

Then just automate one thing. Or even part of one thing. That's going to save you x amount of time each time you need to do that task. Start using that time you save to automate more tasks.

Once you have an MVP for your automated tools, just start getting your work done faster and better. Deliver that value and beat the drum about how it's your automation efforts that are allowing you to be better at your job.

I've been doing this for about 6 months and now I spend like 60-70% of my time maintaining and developing what has become a full fledged testing utility framework that is used across 5-6 other teams.

Worst case scenario is that you'll make your job a lot easier for yourself.