You could say that but it is an odd thing to say. Most people would not express having an up and down day, a tough day, experiencing an emotional roller coaster, as "having mood swings today".
Most of us would say that today is being hard on us in some way. I am having a rough day today, not one of my best. Nothing seems to be going right, I feel like crap, one minute happy, the next like I am in a deep dark hole.
But sure, you can definitely say "I am having (or experiencing) mood swings today". Ya nerd.