Would a Goldie with a brown nose pass on poor pigmentation to any of it's offspring?

Possibly, (does it have black eye rims?), but there are many more important things to consider before breeding a Golden!!
Temperament being one of the most important.

Pigment can come and go with the seasons, pink noses are sometimes called "winter noses", also with age, most puppies have jet black pigment, this does tend to fad as they get older.

The breed standard says "nose preferably black".
We have many Champions in the breed with less than black noses.

Has this Golden had it's hips scored and eye tested?