Yes, you can do that, but you might want to get a little expert advice or do a little library research first.
I don't know what this beam is supporting (one vs two floors, type of roof and snow load, etc), but a fairly safe reinforcement technique might be to rip some 3/4" plywood to 9.5" wide lengths and glue (PL-200) this to one side of the 2X10 beams, then sister another 2X10 to this, sandwiching it, Nail together with #16 spikes in an 8" parallelogram pattern the length of the beam.
But then to the hard work. You will have to take the point at which you want to relocate the posts, break up a 24" X 24" square in the concrete floor centered immediately under the new post location, go down at least 18" (24 would be best) and fill with concrete with cross-pattern #4 (1/2") rebar 6" below the finish grade. Allow this to set at least 14 days before setting the new posts. This may sound like overkill, but trust me, you don't want the new post footing to show stress cracks after 6 months.