Yes you can change a toddlers diaper in public when the toddler needs a diaper change.
Just be mindful of where you actually change the toddlers diaper in public.
Such as do not change the toddlers diaper on a table where people eat food or where someone is eating etc.
If possible though take the toddler to the restroom and change the toddlers diaper there or change the toddlers diaper in your vehicle.
However if you're outside at say a public event and your toddler whether they are 1, 2 or even 3 years old you can lay a changing mat on the ground and change their diaper right there.
Just be sure to cover up the toddlers private parts when you do change the diaper because some perverts could be around.
I've had to change my kids diapers outside when at events such as the fair etc even when they were 3 years old.
I would just lay a changing mat or towel down on the ground and remove their pants and diaper and clean them up as quickly as possible and put a new diaper on.
A lot of people change their babies and toddlers diapers in public even on a sidewalk or picnic table and that's okay but I would recommend doing it on the ground.
The toddler will not be embarrassed at you pulling their pants off and changing their diaper in public because they don't know any better.
When possible I would take the toddler to the bathroom or to the car to change their diaper but sometimes it's not possible.
And the toddler can't stay in a wet or dirty diaper for too long and sometimes you just have to change them whenever and wherever you are.