If your data source is a relational database, there is a good chance that it supports query folding (https://learn.microsoft.com/en-us/power-query/power-query-folding). This means that query steps you define in Power Query will, instead of importing all data and then applying transformations, will instead convert them to your DB's native query language and pass it along so that your DB performs the transformation (e.g. filtering on dates) and then Power Query only loads the (filtered) output rather than all data.
So, you can probably simply make a copy of your pbix, change the filter step (which should be one of your first transform steps), and let your DB only pass the data you need for the period you filtered on. You could even use a parameter to control the filter and not even deal with opening pbix's; just make a copy in the service, change parameter value, refresh.
To address your questions specifically:
What would be the best practice to prevent this kind of scenario from happening?
Probably you can leverage query folding as mentioned above. The page I linked has a pretty good explainer on what transformations fold, how to check if folding is occurring, etc.
Further what is the best approach to only filter the data from the latest quarter?
If you are already going the route of manually creating a new report every quarter, you can probably just manually specify the start and end date of the period you want in a filter on the table.
"best approach" can depend on the situation, but if it were me I would think about just making one report with a Direct Query connection to your database. Way less admin overhead to maintain. https://learn.microsoft.com/en-us/power-bi/connect-data/desktop-use-directquery
Is it possible to only load data into power query or the model from a certain date like having a column with year and quarter as a key. Loading only rows with this key into power query?
Yes, this is exactly the intent of query folding.
Or do I have to create a filter on the dashboard page with a certain date key? But this wouldn't solve the issue that the loaded data size is getting bigger and bigger with time.
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