I'm sure your friends will likely call you on your birthday
In the sentence above, the phrase your friends will likely call you on your birthday is a Complement of the adjective sure. This type of clause is known as a content clause (as opposed to a relative clause, or a comparative clause).
Content clauses are often introduced by the subordinator that:
- I know [that you ate my last chocolate biscuit].
We only use that with declarative content clauses, not interrogative ones or exclamative ones:
- *I wonder that if she is going to the party. (ungrammatical)
- *I saw that how big the elephant was! (ungrammatical)
When to use that in declarative content clauses
We always use that with a content clause when a content clause is the Subject of a sentence:
- [That you were late again] will not impress the powers that be.
- *[You were late again] will not impress the powers that be. (ungrammatical)
We also always use that if the content clause has been moved to a position before the Subject:
- [That I need help] I freely admit.
The sentence above is a version of I freely admit that I need help, where the Complement of admit has been moved to before the Subject, I.
We rarely use that if the content clause is the Complement of a preposition:
- I will see you after you've finished your meeting.
- *I will see you after that you have finished your meeting. (ungrammatical)
[There are a handful of very unusual prepositions such as notwithstanding which allow that.]
In nearly all other cases where the content clause is the Complement of a verb, noun, or adjective the word that is optional. It can be omitted or included as you see fit:
- I know that you ate my biscuits.
- I know you ate my biscuits.
- I'm happy you ate my biscuits
- I am happy that you ate my biscuits.
- The fact you ate my biscuits really gets my goat.
- The fact that you ate my biscuits really gets my goat.
We are far more likely to omit that it's the Complement of a simple high frequency verb, adjective or noun. We are also far less likely to omit that in formal writing:
- The notion you ate my biscuit is laughable. (slightly awkward because of notion)
- We therefore need to underline we going to be there. (awkward because formal and because of the long as well as low frequency verb underline)
In the Original Poster's sentence the content clause is the complement of the simple and high frequency adjective sure. The context is also not formal. The Original Poster can therefore freely omit that. The sentence will be both grammatical and appropriate.
Most of this information is available in: A Student's Introduction to English Grammar Huddleston & Pullum, 2005.