No (or “yes but only for an instant”)

It is reasonable for you to ask this question, since the rules on Ability Score Improvements simply say:

you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can’t increase an ability score above 20 using this feature. -(PHB, p. 115, and other locations)

There is no distinction made in here between Abilities that are natural or reliant on magic, so you might object to the other answers saying that you can only increase from unmodified stats. So let’s assume for a moment that you can increase from the magically increased stat. What would happen?

Consider the following steps in chronological order:

Step 1.) You have an Intelligence of 16 and a Dexterity of 15.
Step 2.) You attune to the Headband of Intellect. Your Intelligence becomes 19.
Step 3.) You get an Ability Score Improvement by hitting level 8, and decide to improve your Intelligence and Dexterity.
Step 4.) Your Intelligence becomes 20, and your Dexterity becomes 16.
Step 5.) The headband’s magic then instantly turns your Intelligence back into 19. After all (DMG, p. 173, emphasis added):

Your Intelligence score is 19 while you wear this headband. It has no effect on you if your Intelligence is already 19 or higher without it.

At step 4 & 5, your Intelligence is not “19 or higher without” the Headband of Intellect: it would be 17 without the headband (previously 16, increased by 1 by your Ability Score Improvement just now). So the magic of the headband would take effect, and your Intelligence “is 19.”

Thus, although the rules (may) permit you to increase an Ability that is already boosted by magic, this particular magic would shift the score (instantly) back.