Probably not, depending on how you interpret the ride skill.
The problem is that after using dimension door "you can't take any other actions until your next turn" (phb pg 221). Abundant step functions exactly like the dimension door spell (phb pg 42). Therefore, a monk cannot take an action after using the abundant step ability.
The easy case is a hostile flier. In order to grab it, you'd have to make a grapple attempt. That would require a touch attack and the grapple itself, both of which are actions. You can't take actions after using dimension door, therefore, you can't make a touch attack or grapple a hostile flier.
For a friendly flier, you need to turn to the ride skill. A kindly DM may treat this as an instance of staying in the saddle (phb pg 80). This allows you to stay in the saddle to avoid your mount doing something unexpected and is not an action. The logic here would be that since dimension door allows you to "arrive exactly at the desired spot" that spot could very well be the saddle.
An alternative view would be that the act of getting in the saddle qualifies as mounting the flying animal. The fast mount ability under the ride skill requires a free action. The DM might allow a fast mount action to be taken in conjuncture with the dimension door so that the action occurs prior to the movement (or in conjuncture with the movement since that is how mounting normally occurs). The kindly DM who lets you use fast mount with dimension door should probably increase the DC for the manuever considering the circumstances.A less generous DM might rule that the mounting roll is made after the movement has occurred, since the check is attempting to determine if you stay in the saddle or not. If that's the case, you wouldn't be able to make a fast mount check since doing so is a free action and dimension door prohibits you from taking any action (even free actions) upon arriving.