Unlikely, since this is June and you harvested it now or in early July. Especially, not in the garage. For long-term storage, pumpkins/winter squash need to be stored below the germination temperatures and above freezing in a cool dry place.
The normal recommendation for storage temperatures is between 40°F and 55°F. However, there is some variation between varieties. Some C. Moschatas can be stored up to 14 months (e.g. Seminole or Neck Pumpkin) and some C. Pepo varieties (e.g. Thelma Acorn Squash) can only be stored for about three months; just to name a couple of squash families.
With some tender love care, you should even be able to get some more fruit from your current vines. or, here in San Antonio (Hardiness Zone 8), you still have time for an entirely new planting depending on the maturity requirements of the pumpkin/winter squash (e.g. 90 days – Autumn Glow or 110 days – Waltham Butternut) variety. Due to high ground temperatures, you may need to start the seed indoors.