The treaty is geared towards international threats, so that would be unlikely.
It could theoretically trigger article 4 of the treaty. The latter states that:
The Parties will consult together whenever, in the opinion of any of them, the territorial integrity, political independence or security of any of the Parties is threatened.
It got used only a few times in history: three times by Turkey (in 2003 on the outbreak of the Iraq war; in June 2012 after the shooting down of a Turkish military jet; and in October 2012 after Syrian attacks on Turkey), then by Poland on the basis of the events in Ukraine, and again by Turkey in 2015 owing to its civil war against IS and PKK rebels.
If the reaction Turkey got in the past two decades is anything to go by, NATO would react similarly to the UN in a similar context: the treaty is about international conflicts first and foremost, in spirit if not explicitly in writing, and states are left to themselves to deal with their internal conflicts.
If things get really out of hands for Spain then article 4 could theoretically result in NATO help. But there's not a shred of sensible evidence that events in Catalonia could blow out of proportion and devolve into violent separatism. Mass protests with occasional violent clashes are a more likely outcome IMO i.e. nothing that anti-riot police can't deal with until a peaceful outcome is negotiated.
It's also worth noting in passing that begging for NATO help to deal with a bunch of poorly equipped rebels is a clearcut case of international humiliation. The UK never sought NATO help when it was dealing with its Irish problems. It would have been the world's laughing stock had it done so. Likewise, as raised by chirlu, for France and its Algerian problems.