Welcome Cinberg!! I don't like small white worms in any of my plants that I bring inside. More often than naught, they came from the soil when you had your Palm outside. If your palm plant soil is part sawdust or sitting on sawdust, sawdust will attract worms, millipeds and centipeds in particular, no question! I have never had to try this but you may want to think about it. Ordinarily, small worms in the soil, flood them out with heavy watering, but a Palm, not so certain that would be the best solution. You could try a weak solution of bleach in your water for your Palm, but here is what I would do, having just repotted all my "tropicals"! I would bang the pot a few times on different sides to loosen the soil (providing it is not 10' high (LOL), I would get rid of that soil outside somewhere to fill garden beds, whatever, and I would carefully wash all the soil out of the roots of that Palm, all of it, even with a garden hose if necessary. First off, you will be able to see if your Palm is root bound? Secondly, you will be able to get rid of any or many metals and oxidates that may have overaccumulated in the soil over time, and finally, I would repot it in newly purchased potting soil, and of course a good watering when first doing so. Let's recap: Ruled out "root bound" scenario, ruled out "accumulation of metals and oxides over the years", ruled out worm infestation - hey, the Mayo Clinic couldn't help you any further! LOL. Give it a try!! Cheers!
Great for the weekend "Palm Springs Spring Break" party or luau, but then dump it with the rest of the decorations. A "Martha plant" if there ever was one. 
