You do not need an appointment for individual portrait sessions. Only large groups require an appointment.
On Mercer: be aware that the Arboretum is on the west side of Aldine Westfield Rd while the Botanical Gardens are on the east side. If you are meeting folks there it is important to note the difference. Both are free, but donations are welcomed. The Botanical Gardens were thick with portrait shooters this past weekend, weekdays are much more open. They close at 5pm every day.
The Botanical Gardens are much more popular with portrait shooters for obvious reasons, and altho when I'm there I'm usually doing macro work, I have made mental notes of spots for portraits for the few times I've done that. The Arboretum is open park area, trees (duh) and grassy expanses mostly. There is/was a wooden deck/bridge near a cypress bog that appeared to have potential.
In the Botanical Gardens the pond in the back half of the park is suitable, the gazebo and/or floating deck are popular spots. I've seen folks literally lined up for the deck.... There are benches and some grassy areas that are useful.
In the front of the Botanical the main areas are the stone archway beside the (former!) daylily area and the adjacent bridge over the lilypond (behind the greenhouse as you enter, slightly to your left). Near there is the water curtain, that's always popular but full sun may be an issue and there may be people in the background to deal with. Along the far left side of the frontal area is another waterway with hills and shade. This is actually my favorite area as the shade is softer and the hills provide some cleaner backdrops (sans people..) Near the "house on stilts" is a small water feature with a nice swing. On weekends it is very popular.
I would recommend you scout the park ahead of time, being mindful of the sun and shadows. Google earth can help.
Mercer Arboretum & Botanic Gardens