If a contractor did not finish the work that was to be completed per the quote given can the contractor file a lien? They are demanding to be paid but refuse to come back and actually complete the job.
It's generally possible for a contractor to file a mechanics lien before the project is complete, yes. However, that lien claim would only be appropriate to the extent that the contractor has performed work but not been paid for their work. So, if some work was done and not paid for, a contractor may be able to file a lien for that work. But, a valid and enforceable lien claim cannot be filed for amounts representing work that's not actually been done.
These articles should be useful to you: (1) I Just Received a Mechanics Lien Threat – What Should I Do Now?; and (2) A Mechanics Lien Was Filed on My Property – What Do I Do Now?
If the contractor argues that he or she has not been paid on the labor provided or materials furnished on the project, an Affidavit of Mechanic's Lien can be filed. Whether that lien is valid and the contractor is in fact entitled to payment will likely be part of a breach of contract claim. After receiving the Mechanic's Lien, the owner can file a Notice to Commence Suit giving the contractor 60 days to sue on the lien or lose those lien rights for good.
The contractor could lien the property for the work the contractor has already performed, if the contractor has not been paid for some or all of that work. The contractor cannot record a lien for future work that has yet to be performed.