Do, yes. Usefully, less so
First, if you are incapacitated the companion is free to do anything which it already could be commanded to using a bonus action. So, for this to be useful it would need to occupy less than that part of your action economy. We're also fully trusting your companion's judgement, and given our methods, it's might be better than our character's.
Let's also cover some of the broader issues and limitations. The incapacitated condition ends concentration, that means we won't be able to banish ourselves, nor make ourselves laugh uncontrollably. Since we want to do it on our own, we can't rely on anyone else's concentration either, and we can't alleviate it by methods which avoid concentration, as that wouldn't let us end it at will either.
Sleep it away
One method to do this requires a bit of set up. Specifically we want a number of spell glyphs (only 200 gp each) which are storing the sleep spell and which are set up such that we can trigger them with our movement or free object interaction. Then, huff some burnt othur fumes (DMG p. 258). You can succeed the initial constitution save, though the damage will make it easier for the sleep spells to affect you. Our loop is as follows, once you are done having you turn, trigger a spell glyph and have the sleep spell affect you (if you're not alone, make sure you have the least hit points or that there's enough juice in the spell to also affect you). You will then fall unconscious and be incapacitated. Assuming everyone (including your companion) leaves you alone, you'll stay that way until the start of your turn when you fail your saving throw against the othur fumes you inhaled and wake up from the 1d6 damage, free to take your turn.
Embrace your inner pot plant
A wilder, and more
fate dice dependent method is to be a Wild Magic Sorcerer and make sure you cast your leveled spell at the end of your turn. Prior to that spell you have used Tides of Chaos and so get a wild magic surge in order to get it back (thanks DM!). On that surge you roll 41 or 42 so that:
You turn into a potted plant until the start of your next turn. While a plant, you are incapacitated and have vulnerability to all damage. If you drop to 0 hit points, your pot breaks, and your form reverts.
While by default a 1 in 50 chance, you can improve those odds by being a 14th level sorcerer for Controlled Chaos, and or rolling 81 or 82 on the surge to take another action (casting a new spell triggering a surge). Another method to improve your odds would be to be in a wild magic zone, assuming that allows multiple surges on the same spellcasting.
As a footnote, I'll mention that the 'take me away while it's not my turn' spell, blink does not work because it leaves non-incapacitated on the ethereal plane, nor does the 9th level spell imprisonment work as the repeatability is limited and it has a casting time of one minute.