It means that the quality is excellent or "first class" (i.e. of the best quality or in the highest division of quality).

While I haven't found a source for "first quality", there is a defintion for "first quality workmanship" in US Legal definitions:

first quality workmanship means construction which is free from any defect materially affecting appearance or serviceability.

It means that the quality is excellent or "first class" (i.e. of the best quality or in the highest division of quality).

While I haven't found a source for "first quality", there is a defintion for "first quality workmanship" in US Legal definitions:

first quality workmanship means construction which is free from any defect materially affecting appearance or serviceability.

You ask specifically:

is it correct to say like this?
Starbucks is famous for its first quality coffee.

It is grammatically correct, but I would prefer to say:

Starbucks is famous for its excellent coffee.

It means that the quality is "first class" (i.e. of the best quality or in the highest division of quality).

It means that the quality is excellent or "first class" (i.e. of the best quality or in the highest division of quality).

While I haven't found a source for "first quality", there is a defintion for "first quality workmanship" in US Legal definitions:

first quality workmanship means construction which is free from any defect materially affecting appearance or serviceability.