The short answer is no, we have no reason to believe that the historical Laozi traveled to India and tutored the historical Buddha. This is a folk tradition that emerged during the period of Buddhist-Daoist syncretism in China, many centuries after the supposed lifetime of Gautama Buddha, and wrongly projects the very real interrelationships between the two faiths that developed within China in that later period all the way back to their founding figures. There are a number of reasons not to accept this tradition, which I'll describe below.

One, the controversies surrounding Laozi (老子 "the Old Master") as a historical figure and the Daodejing as a historical work suggest that he may not have lived in the time of the Buddha, or even have been a real individual at all. We have many conflicting accounts of Laozi's identity and life, none of which are considered to be particularly reliable. Was the real Laozi a man from Chu named Li Dan, as Sima Qian claimed? Or was he really Taishi Dan, or maybe he was the same person as Laolaizi or Li Er, mistakenly split off by Han historians? Was he a composite of all of them? Depending on the answer, he may have lived anytime from the late 6th century BC (about contemporary with Gautama Buddha and Confucius) up to perhaps the mid 4th century (well after them). The oldest known editions of the Daodejing, recovered from tombs of the Warring States period, date only to the late 4th century BC yet differ significantly from the received text, which tells us that the text evolved over time under the pen of numerous authors. So in short, it is not at all certain that there was any one person who bore the title Laozi and created the early Daodejing himself, much less that he was contemporary with Gautama Budda, much less that they knew each other.

Two, the time of the historical Buddha was not one in which we have firm evidence of travel, trade or communication between China and India. The first direct reference to India in the Chinese record is the report of the 2nd century BC emissary, Zhang Qian, who traveled Central Asia and learned of the Indus River region. It's not until the 5th century AD that we start to get accounts of travel between China and India, mostly among Buddhist priests- either Indians going to China to proselytize or Chinese going to India for study or to acquire scriptures. The hardships of travel over such distances, as well as the difficulty of translating complex philosophical discourse between unrelated languages, are apparent, even in this period where Silk Road trade meant the routes were well-known and reasonably safe. So the folk tale of a single wizened old man making that trek a thousand years earlier, to an unknown and unmapped land, and then somehow becoming the tutor to a foreign prince is highly improbable.

Three, there is no indication that Buddhism and Daoism influenced each other prior to Buddhism's rise within China from about the 1st century AD onwards. If Laozi was a real person who did espouse a worldview like the one captured in early editions of the Daodejing, and he played an important role in the education of Gautama Buddha, then we should expect to see, if not necessarily agreement with Chinese Daoist doctrines, then at least some kind of engagement with them in the early Buddhist tradition, but we find nothing of the kind. To give an impression of the general incompatibility between early Buddhism (and, by extension, Indian thought more generally) and early Daoism (and Chinese thought broadly), let's look at their core ideas around death and physical existence: Buddhism posited a cycle of rebirth fueled by karma, and therefore a supreme spiritual objective of escaping the cycle. The material world was samsara, an endless wheel of dissatisfaction and suffering. Daoist doctrine was nearly the opposite; the physical world is inherently good, and the forces of nature serve as the model for human life and the path to spiritual attainment. They had no doctrine of rebirth, and assumed that death meant becoming a ghost bonded to their still-living kin- and their supreme goal was not to escape the world but to prolong and enhance their existence within it, at the low end through practices for health and longevity and at the high end, through the quest for immortality. Each has a great deal in common with other traditions from their respective regions but virtually no similarities with each other.