In theory, a quadrupedal Khajiit could give birth to bipedal offspring, but this may never actually happen.
The Lunar Lattice
From the book The Moon Cats and their Dance in The Elder Scrolls Online:
The hairless scholar peered at the Clan Mother through little windows on his nose and said, “I want to know about your different breeds. Is it true that what phase of the moons you are born under determines your physical morphology?”
And the Clan Mother said, “Indeed, hairless scholar. I was born as Jode was waxing and Jone was new, so I am an Omhes-raht. My daughter here was born when Jode was waxing and Jone was full, so she is a Senche-raht. Thus we are nothing alike.”
The page on Khajiit physiology at The Imperial Library says that the form a Khajiit takes depends entirely on the phases of the moons at the time of their birth. As such, children may look very little like their parents, depending on when they were born.
(Jode is apparently Secunda, the smaller moon, and Jone is Masser, the larger moon.)
The text seems unambiguous: if an Alfiq (housecat) were to give birth while Secunda is waxing and Masser is full, the child would be a Senche-raht, an enormous cat that would grow to the weight of fifty Altmer. But this particular parentage seems implausible. I am not aware of any canon explanation for why births don’t seem to be evenly distributed between all forms of Khajiit, but I can think of a two possible explanations. Both theories tie back to the phases of the moons in some way.
As far as I know, the only way to avoid the fate given by the Lunar Lattice is if births do not happen during implausible phases of the moons.
Khajiit are only fertile during certain lunar phases
Perhaps the Khajiit are only in heat during certain lunar phases. (In the real world, this apparently does not happen, although there are persistent rumors of fertility in humans being tied to the lunar cycle.) If a Khajiit's gestation period is very predictable, then a conception during particular lunar phases would result in the child’s birth during other predictable lunar phases. An Alfiq may only be in heat during lunar phases that would result in her children also being Alfiq, or at least small varieties of Khajiit. Regional differences in Khajiit population could be explained either by the genetics of local populations or influences of the local environment on the Khajiit's estrous cycles. (This assumes that Khajiit have something analogous to the estrous cycles of real-world cats.)
Under this theory, an Alfiq would never be in heat during a time that would result in a birth during a waxing Secunda and full Masser, so an Alfiq would never give birth to a Senche-raht.
Khajiit can only go into labor during the appropriate lunar phases
If a Khajiit is pregnant with a Suthay-raht, perhaps she can only go into labor when Secunda is waning and Masser is new: the expected lunar phases for the birth of a Suthay-raht. Under this theory, each form of Khajiit would be predisposed to conceive only certain plausible types of Khajiit. The mother would go into labor only when the moons are in the appropriate phases. If labor were induced early, perhaps either the child would be stillborn, or the magical effect of the lunar lattice would cause the Khajiit’s form to change. An Alfiq would not conceive a Senche-raht, so there would be no births in an Alfiq population when Secunda is waxing and Masser is full.
Unfortunately, none of this says whether a quadrupedal Khajiit could give birth to bipedal offspring. But if this has never been observed to happen, then the theories above could explain why it doesn't happen, without contradicting the Lunar Lattice. We know from the text above that a bipedal Khajiit (Ohmes-raht) can give birth to a quadrupedal Khajiit (Senche-raht). My suspicion is that the larger quadrupedal Khajiit at least occasionally give birth to bipedal Khajiit, but I am not aware of any canon examples.