Can I put a 20Amp breaker on my existing 15AMP garage circuit?
Don't try to run a saw and a dust collector on the same circuit

You seem to think a 20A circuit will magically support both. No, it won't. You sink the time, energy and cost into getting that single 20A circuit, plug both in, and you'll get a snap for your trouble.  Stop trying. 

You need a 15A circuit for the dust collector, and Bob's your uncle.   For the table saw, a [different] 15A circuit might work.   If the saw has a NEMA 5-20 plug (rotated neutral), then  you definitely need a 20A circuit including 20A wire.    

Now if the saw has a standard plug but still trips a 15A breaker reasonably often, and you are prepare to dedicate that circuit to this saw only, then take a picture of the motor nameplate on the saw, and post it here in a new question asking "Based on this motor nameplate, can I bump this breaker? Canada code." There might be a magical exception.  Maybe.  There are some numbers that need to be crunched.  Don't go slapping in a 20A breaker on that hope. Capische? 

A 20A circuit absolutely requires #12 copper or #10 aluminum

You cannot simply slap a 20A breaker on any cable that uses #14 copper or #12 aluminum.   Remember, even if  you are able to exploit that magical exception, that only works on a circuit dedicated to one specific tool. 

You can tell what  your cable is by reading the words printed on the cables for about 1 foot.  There's a lot there, but there should be something like 14/3 AWG or 14 AWG, and Copper, CU, Aluminum, or AL.  

Given that the cable near the panel is in unfinished space, and garages often are too, you may be able to run new cables and then obtain new 20A breakers for those cables.  This is the normal approach when acquiring new power tools.  It's part of the ritual. 

Check with your local EATON dealer (or ThreePhaseEel if you can post a picture of your panel's label), but it appears at first glance that your panel takes EATON type BQL breakers (or BQLT if you need double-stuffs).  These cost C$20 per space at Lowes, which figuring for exchange rate and VAT is probably about twice a common BR breaker.  They bolt down, so you must power down the whole panel to fit them.  But that's easy enough. 

"But Commander panels are dangerous!"

OK, let's think this through.  There are dangerous panels out there, but their troubles generally fall into 3 different areas.  


The bus stabs not having good contact, arcing and starting fires. This panel uses bolt-down breakers, which makes this a complete non-issue.  
The neutral and ground bars making poor contact with the wires for some reason.  Later research proved much of this trouble was actually caused by screws not being torqued down to spec.  Because of this, US NEC now requires installations use a torque screwdriver.  In panels with suspect G/N bar problems, this is the best answer. 
The breakers themselves being duff.  FPE and Zinsco got caught out cheating their UL listings.  I don't know why CSA (Canada's UL) delisted Commander panels, but they did. The fix here is replace the breakers with reliable modern stock. That's not an option for FPE or Zinsco (the $80 Connecticut Electric replacement breakers are even worse than the originals)... but Eaton's BQL line is the replacement for Challenger, and it appears to be simply a repackaged BR breaker with a  boltdown flange instead of a stab clip. And, they are sanely priced.