Are you referring to Google Ads attribution model? The default one used is Last Click as per here:
Last click: Gives all credit for the conversion to the last-clicked ad and corresponding keyword.
As of the most recent news, Google Ads is removing First click etc attribution models as per here:
[…] first click, linear, time decay, and position-based attribution models across Google Ads and Google Analytics 4 will be going away. Data-driven attribution, last click, and external attribution won’t be impacted.
Starting in June 2023, we’ll remove the ability to select first click, linear, time decay, and position-based attribution models for conversion actions in Google Ads that do not already use one of these models. Then, starting in September 2023, we’ll switch any conversion actions still using these models to data-driven attribution.
I’ve also opened a topic on Matomo forum about it here.
I believe your discrepancies that you are seeing might be due to different attribution models used: in Matomo you might have a different one than the one used in Google Ads.
What you could do is do a comparison apples-with-apples, test on a very low cost $ Adwords ad the clicks and conversions, and in real time check what gets tracked; of course Adwords might have a delay so if that is the case, you can check a few hrs later or next day; but I’d setup a test like this, even on a demo test product or a URL you don’t get many sales on to really see the conversion(s) for it. This way you could do a first-click attribution, or do last-click attribution, test both ways and see the difference between Matomo and Adwords.
My preliminary hutch is that Adwords is using last-click model and your Matomo is using a different attribution model; you are seeing same conversions but different attribution models.
Please let us know how you go with this as there might be other Matomo users in the same situation like yourself.