Use the UPSELL feature for this.  Also works the same way for Odoo 17.

Example Product:

Create the Subscription Sales Order, confirm it, then change the Date of Next Invoice:

Note: At this point, you need to change the Quantity on the Sales Order to 0, because Odoo will automatically increment it when you use the Upsell feature:

Click Upsell:

This creates a Quotation with a pro-rated amount based on the difference, with a discount automatically calculated for the correct percentage:

Once this is confirmed, change the Quantity to 1

You can then Invoice your Customer for the partial amount due this quarter.

In this case, they are getting a discount (automatically calculated by Odoo) for the first 25 days of the quarter they didn't receive the service - 25/[31+29+31] = 27.47%.

Invoice:

You can check that the quantity on the original subscription Sales Order is 1 (incremented automatically from 0 by Odoo when you created the upsell) because you are going to invoice 1 full subscription at the start of the following month.