They have somehow shown their supports and encouraged me to pursue postgraduate study.
Right collocation (v. show, n. support), wrong form (at least in this context).
‘Support‘ is one of those annoying words that can be countable and can be uncountable. In its countable form it refers to a physical support (or supports), for example the supports used to stop a building from falling down:
In its uncountable form, ‘support‘ refers to a more abstract support that may be physical but can also be emotional. I think it was this second meaning that you were aiming to communicate:
- They have somehow shown their support and encouraged me to pursue postgraduate study.
Again, the collocation is good: v. show, n. support!