Ok, I'm wondering . If Uchiha and Uzumaki have a child together, there are possibility that the child inherited Sharingan right? Since Uzumaki have ties with Senju then, can the child awaken Rinnegan?
No, just an average/general Uchiha and Uzumaki marriage cannot give birth to a child who is capable of activating Rinnegan. In order to activate Rinnegan one requires chakra of both Indra and Ashura which enables Hagoromo's chakra. Please refer the below url. http://naruto.wikia.com/wiki/Rinnegan
The correct answer is IT DEPENDS.
Rinnegan, as multiple users have pointed out, can be obtained by combining chakra from Indra and Asura. In each generation, one descendant of the brothers inherits their Chakra.
Thus iff the "chosen" descendents of both Indra and Asura have a child together, their offspring would inherit both Chakras and would have a Rinnegan naturally. Since both Uchihas and Uzumakis have the potential to inherit the Chakra from Indra and Asura, it MAY be possible for an offspring to inherit Rinnegan naturally.
Disclaimer: The above scenario is hypothetical and is intended to point out that Natural Rinnegan may be a possibility. However, its highly likely that all "chosen" successors are male from what we've seen.
Madara Uchiha did state that you need both Senju and Uchiha chakra to awaken the Rinnegan, but it's actually the fusing of Indra and Ashura's chakra that awakens a Rinnegan.
According to the wiki:
It is possible to awaken a new Rinnegan by obtaining Hagoromo's chakra, either by combining the chakra of his sons, Indra and Asura, to recreate his own, or by directly receiving it from Hagoromo himself.
As for the Sharingan, there is a high chance of the child awakening it, as Sarada, Sasuke's daughter (Half Uchiha and Half Unspecified Clan w/ Unspecific Abilities) still managed to awaken the Sharingan. Also because the Uzumaki Clan and Uchiha Clan are distantly related by blood (no longer closely because the clans previously continued to expaned, causing blood ties to become slimmer). But, there is still a high chance of the child getting the Rinnegan.
This child would most likely be very powerful as they have two of the strongest clans blood running in them!
The Rinnegan is a mutation of Otsutsuki chakra and nature chakra which explains Kaguya's own Rinne Sharingan on her forehead. Hagoromo is more of the watered-down version of her powers and prowess on it, just continue to go down even more so through generation after generation.
Now if an Uchiha and an Uzumaki had a child, then yes it is possible. Moreover, if that child unlocks the Mangekyo Sharingan, the side effects will also be negated due to the dense chakra of the Uzumaki which you can compare to Obito's situation with the Senju DNA.
All Uchiha have Indra's chakra and all Senju (Uzumaki related) have Ashura's chakra. Now, Madara/Sasuke and Hashirama/Naruto are special because Indra's and Ashura's reincarnated into them.
To answer the question, yes, the child will be able to obtain Rinnegan. He will be most likely like the Sage of the Six Paths before he obtained Sharingan and Rinnegan.
The sage wasn't born with Rinnegan. He started with nothing, his eyes were almost like a Byakugan. When he lost his lover, he obtained the Sharingan. Afterward, learned how to control sage power and went fight his mother. His mother made him kill his brother (he healed him a few moments after, ain't got time to explain all) which allowed him to awaken the EMS. He had the EMS and sage powers and he ended up with the Rinnegan.
Uchiha possess the Sharingan. Senju possess the Sage Chakra.
Of course, if they have a child, he will be able to obtain the Rinnegan. Although he will have to overcome hardships like losing a friend (Sharingan) a lover (MS) and take sibling eyes (EMS) and since he already has Sage Chakra from Senju's parents, the Rinnegan will be awakened.