The cactus is a background object which appears in Don't Hug Me I'm Scared 1, 2, 5, and 6.
In episode 1, it can be found on the windowsill and has a face made of needles stuck into it.
In episode 2- TIME, it can be seen next to Roy's computer and It is also seen in Red Guy's bedroom on the right.
In episode 5, it is on the shelf (along with the radio and hat)
In episode 6, it is seen on the windowsill on the left in the bedroom.
It is similar to the hat and radio since they also appear in the background of episodes.