Yes, some believed in a Round Earth, but we don't know if they tried to measure it.
I'm going to start from the basis of the Round Earth concept; Hellenistic Astronomy was based upon Babylonian astronomy (which was more than likely taken from Sumerian astronomy, the earliest civilization, so far as we know). Greek astronomers like Pythagoras, Aristotle, and Eratosthenes developed their theories based upon the observed science that came before them through Babylon, and much of what we attribute to Hellenism should actually be owed to the Babylonian and Sumerian civilizations. These Mesopotamian civilizations did not believe in a flat Earth concept. (the apple fell far from the tree). see: Babylonian Astronomy
In Babylonian cosmology, the Earth and the heavens were depicted as a
"spatial whole, even one of round shape" with references to "the
circumference of heaven and earth" and "the totality of heaven and
earth". Their worldview was not exactly geocentric either. The idea of
geocentrism, where the center of the Earth is the exact center of the
universe, did not yet exist in Babylonian cosmology, but was
established later by the Greek philosopher Aristotle's On the Heavens.
In contrast, Babylonian cosmology suggested that the cosmos revolved
around circularly with the heavens and the earth being equal and
joined as a whole. The Babylonians and their predecessors, the
Sumerians, also believed in a plurality of heavens and earths. This
idea dates back to Sumerian incantations of the 2nd millennium BC,
which refers to there being seven heavens and seven earths, linked
possibly chronologically to the creation by seven generations of
With that being stated, it's not impossible to believe that these views spread in some form to the Indus Valley and from there to the Shang Dynasty (cultural diffusion), given the span of a few thousand years. Therefore, the Round Earth view more than likely existed, not as the dominant ideology, but as an emergent one under scrutiny that was absorbed. Perhaps some ancient Chinese astronomers believed in a Round Earth, but like the Greeks, that knowledge was lost and a flat Earth concept became the dominant view through syncretism.
What we do know is that Yu Xi was the first Chinese astronomer documented to challenge the flat Earth concept in China. He held the basis of a heliocentric model but did not expand upon that, since he had to challenge the concept of a spherical Earth and spherical bodies in the heavens.
Extract from Yu Xi's wiki:
Yu Xi wrote a critical analysis of the huntian (渾天) theory of the
celestial sphere, arguing that the heavens surrounding the earth were
infinite and motionless. He advanced the idea that the shape of the
earth was either square or round, but that it had to correspond to the
shape of the heavens enveloping it. The huntian theory, as mentioned
by Luoxia Hong (fl. 140-104 BC) and fully described by the Eastern-Han
scholar-official Zhang Heng (78-139 AD), insisted that the heavens
were spherical and that the earth was like an egg yolk at its
center. Yu Xi's ideas about the infinity of outer space seem to
echo Zhang's ideas of endless space even beyond the celestial sphere.
Although mainstream Chinese science before European influence in the
17th century surmised that the Earth was flat and square-shaped, some
scholars, such as Yuan-era mathematician Li Ye (1192-1279 AD),
proposed the idea that it was spherical like the heavens.
This expands upon what was shared in the other answer. Some ancient Chinese scholars believed in a spherical Earth despite the ideas not being popular to the mainstream beliefs of their time.