@elise_williams Butler doesn't have access to the Google Calendar API directly, but you could create a button that sends an email (so long as you're on a paid Trello plan) into Zapier email parser that would then extract the data and create an event in Calendar.
This won't search for availability though. For that, you need to have something written that has access to your Calendar API. Google Apps Script is the best way to do that.
I have created a Trellinator package that I use internally with my clients to schedule recurring meetings.
Currently it works from Trello card descriptions that are formatted like this:
Emails: person1@domain.com,person2@domain2.com
Days: Monday,Tuesday,Wednesday
Times: 11:30,14:30
Location: https://zoom.us/j/MEETINGID
Duration: 30
The Trellinator runs as a G-Suite user, so as long as person1@domain.com and person2@domain2.com are both on Google Calendar and the Trellinator user has free/busy access (at least) to those calendars, it will look for a spot with shared availability and automattically schedule it in.
In the above example, it would look for a time on those days, in that time range, for a shared slot of 30 minute availability.
You can also set more than 2 people, so for example if you need to regularly schedule meetings with 8 different people in various configurations, on a monthly or weekly basis, this is what it's for.
The same package could be reconfigured so that, for example, you request a meeting on an ad-hoc basis with one or more people by creating a new card in a "Meetings" Trello board and it would find the next spot you were all available and put it in the calendar, or that it would only search availability in YOUR calendar to undertake a specific task, for example you add a due date to a card with a custom field value called "Duration" and it will then look for availability in your calendar to get that done by that date and if it can't find enough time will warn you that you can't do it.
Usually I post code examples when I share Trellinator solutions, but this is quite complicated and the code isn't open source. However you can find out more about Trellinator here:
https://community.atlassian.com/t5/Marketplace-Apps-articles/Introducing-Trellinator-Automate-Trello-with-Google-Apps-Script/ba-p/925271
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