Yes, you can do this the way you've proposed and actually I think it's a perfectly reasonable outside-the-box solution. What you'll end up with is not that different in principle to how some Mid Century Modern pieces have their round legs attached.
One caveat though, I do think you'd be better off using two dowels if you plan to lean on the table much when it's in use. A single dowel placed centrally will allow for a lot of twisting or warping of the top (far more than you'd guess, even if the top is very thick and stiff) so adding two legs would be much better.
If you're set on using just one leg mount it off-centre, toward the side you plan to sit at.
Is there a piece like this photo, either square or round
You'd need to make this but it's a simple enough operation, just requiring cutting the wood to size and drilling a suitable hole through it.
If this joint were described in woodworking guides I bet it would say that ideally you'd want to create a shoulder on the dowel (essentially forming a circular tenon on its end, the drilled hole in the block then becoming a round mortise) but given the way this will be mounted I think that's complete overkill. With a good tight fit* between the dowel and the drilled hole you'll get an excellent glue join, and once the block is firmly attached to the tabletop from underneath the leg isn't going anywhere.
which I can glue to the board
Don't glue the board to the tabletop, use screws at each corner instead. Pre-drill and countersink as needed, see previous Answer for details if required.
I want to place a 1.25" dowel
You didn't specify the wood type here, but I would recommend you go with a hardwood rather than a softwood especially if you go with just one leg.
In either case it's a good idea to select a dowel with straight grain and minimal runout (grain should be parallel to the sides, not at an angle) to be as stable as possible, but this is most critical if you go with softwood. And avoid picking a piece that has a knot in it.
*Go for a tight fit but in practice you might not find this possible, as the dowel may be a fraction smaller than the available bit despite them being nominally the same diameter. If this does occur use epoxy to glue the pieces together, almost all regular wood glues (including all PVA types and most polyurethanes) have very poor gap-filling properties.