You must slacken off both the large nut on the brake stub axle
and the axle by the speedo drive gearbox then turn each chain adjuster nut by the same number of flats (this assumes the rear wheel is in line with front). Do not slacken either the large nut or the axle too much, both should remain "nipped up" so turning the adjuster nuts takes a little effort. When the chain is adjusted correctly, remember to retighten both the axle and the brake stub axle nut.
Regrettably, specifically the 1965 Triumph 350/500 Owner's Handbook is not available new (although you might find a used copy on Ebay); however, there are few differences from either 1964 or 1966, both of which are available new from the publisher -
Andover Norton; either will detail basic maintenance for you.
Undo the axle by the speedo drive gearbox and pull it out. There should be a "cotton reel" shaped spacer between the large nut against the speedo drive gearbox and the chain adjuster; if the spacer does not drop out by itself, pull it out; you now have space to pull the wheel towards the right hand side of the swinging arm, disengaging the splines on the wheel hub from those in the brake drum.
Hopefully, the splines will have been greased with something tougher than ordinary grease - ideally boat trailer wheel bearing grease. However, if so, be aware boat trailer wheel bearing grease is
very sticky - if you get dust, grit, etc on it, the only way to get it off is degrease, removing all the existing grease (and foreign objects) and regrease with new.
When I ran a bike with a q.d. hub, having discovered the above the hard way, I used to carry a plastic bag to put over the hub splines and enough cable ties to secure the bag there, for if ever the wheel had to be removed for any reason.