Air carrier Captain reported smoke in the cockpit and seating area of the aircraft after adjusting the temperature controls. The crew diverted to a nearby airport for a safe landing and the passengers were removed from the aircraft.

Narrative

Several minutes from my top of descent my First Officer checked the cabin temperature and mentioned that it was in the low 60s and we began turning both cabin and flight deck temperature control knobs to increase airplane temperature. We had been running the temperature as cold as we could due to the summer heat. The knobs were then both turned from a max cold position to a modest 9 o-clock position. Shortly after increasing the control knobs; airduct temp began reading a drastic increase; approximately 10 degrees Celsius every few seconds. Air temperature coming through the ducts was noticeably warmer very quickly. I began adjusting the knobs to a cooler setting and then the Cabin Pack Hot caution light illuminated along with appropriate warning chime. This happened within about 10 to 15 seconds of first adjusting the knobs. The checklist was performed and shortly after we received a call from the Flight Attendants informing us that we had smoke in the Cabin. Right as they informed us I noticed we had smoke in the Flight Deck as well. Both First Officer and I noticed a distinct smell with a noticeable haze forming. I quickly discontinued the call with the Flight Attendants and both me and my F.O. called 'Smoke Don Masks' and we quickly followed Mask donning procedures. After communication was reestablished with my F.O. I began running the IAC (Initial Action Checklist) for Smoke (In Flight) while my F.O. continued to fly the airplane since he was already pilot flying. While performing the IAC checklist I made a quick radio call to ATC to inform them that we had smoke in the aircraft and we [required priority handling]. I then gave radio control over to my First Officer to finish the checklist; perform an Approach Checklist; and coordinate with the flight attendants on what to expect when landing. They requested that I make a P.A. to the passengers. Before making a P.A. to the passengers; I checked to see how my F.O. was doing. He was set up for a visual backed up with an ILS into ZZZ for XYL and was direct ZZZZZ. By this point the smoke was starting to clear out and was barely discernable in the Flight Deck. I called back to the Flight Attendants and asked if they were seeing the same in the Cabin and they both agreed that the smoke wasn't as bad but still noticeable. My First Officer continued to fly the airplane to landing were we taxied into the gate and met with emergency personnel. Normal deplaning was conducted and no limitations were exceeded to the best of my knowledge. Donning my mask felt like it took longer than it should have since the controls for them are rarely used or observed. It had been more than 6 months since my last CQ event and having better familiarity with the various controls could have saved me a few seconds with the donning procedure.