NO, or at least not until about 900 years after both Buddhabhadra and Bodhidharma died.
In Meir Shahar's The Shaolin Monastery: History, Religion, and the Chinese Martial Arts, there is a whole chapter (number six) dedicated to that topic. It basically debunks this (with reference to studies and historical sources) as a myth that had been raised as late as the 17th century and later. The conclusion ends as follows:
The seventeenth-century transformation of bare-handed fighting was accompanied by the emergence of a novel martial arts mythology. Shaolin monks
gradually accepted a legend that had originated outside the monastery according to which their fighting techniques had been invented by the Chan patriarch Bodhidharma. The legend of the Buddhist saint evolved in conjunction
with the myth of a Daoist immortal. As the “External” Shaolin martial arts
were attributed to a Buddhist master who reputedly resided on the sacred Mt.
Song, an “Internal” school of fighting was ascribed to a Daoist recluse who supposedly hid on the holy Mt. Wudang. The two legends matched each other in a
perfectly harmonious mythological structure, the flawless symmetry of which
has likely been the source of their ongoing appeal. (p. 181, emphasis mine)
The main point is that is only in modern times (17th century and later) that stories were made up to provide a unity of religion and martial arts that provided further credibility and myth to the alleged superiority of (Shaolin-)Kung Fu. This mixes religious aspects of trying to found legitimization on lineage with martial traditions of questionable historical lineage. Let me handle both separately.
On Zen Buddhism, Buddha, and the Shaoilin monastery
The thought that legitimization has to be based on tracing one's lineage to Bodhidharma is based in Chan (Zen) Buddhism as such:
The eighth century witnessed the flowering of a new school of Chinese Buddhism [...]. One of the Chan School's novel traits was the belief that the truth revealed by the Buddha could be directly transmitted from master to disciple. At least in theory, it was no longer necessary to study the scriptures. Instead, the unmediated mind-dharma (xinfa) could be handed from teacher to student. To legitimize this claim, Chan masters had to show that their mind-dharma had been transmitted to them through a lineage going back all the way to the Buddha himself.
The monastery itself was most certainly founded by an indian-born monk "in the last decade of the fifth century" (p. 9), and even though it is widely accepted that Buddha was from India and at some point (about 40 years later) was near the location of the Shaolin monastery, it is only through the 6th to 8th century that the Shaolin monastery and Mt. Song have been firmly linked to the alluded Chan patriarch. In truth, neither him having founded this strand of Buddhism nor him having taught religion or exercises at Shaolin monastery can be considered anything but a myth as it does not add up with historical facts and is only to be found in later records contradicting the original ones (see p. 13).
Buddhist monks and martial arts
In general, the Buddhist faith categorically forbids violence. There are archaeological proofs that the Shaolin Monastery in particular did fight in 610 and 621, though, which secured a lasting protection of their status as a Buddhist monastery with certain property rights by the dynasty they helped to establish. That is, they fought in violation of their general faith in a time where many Bhuddist monasteries where sacked and disbanded, probably to save themselves. The existence of Buddhist war gods, the appointment of a Shaolin monk as a general in the imperial army after the win in 621, and some textual evidence suggest that there used to be martial training at the monastery at that time. This was probably practiced in the perimeter rather than the monastery proper where a more pure Bhuddist lifestyle of non-violence and vegetarianism was the norm (see chapter 2 for all this).
The actual famed "Shaolin Kung Fu" cannot be traced to any sources earlier than about 1600 AD, though (see chapter 3). The monastery used to be famous for its staff fighting rather than bare-handed fighting before that. The latter apparently developed or was brought there in the 17th century. This puts the claim that either Indian monk (the founder or Bodhidharma) actually had any part in the development of the systemized martial art that is known as Shaolin kung fu even more into question.
It is a common myth which perpetuated for reasons of religion and, especially in the 20th century, advertisement and tourism. The Shaolin monastery is set at a very important strategic point near a local capital and hence had been linked to wars and consequently martial arts since shortly after its foundation. It cannot be ruled out that Kalaripayattu and Shaolin kung fu are related in some way insofar as they share principles and informed each other, but certainly not as a direct lineage from India to China originating in the 5th or 6th century. The trademark bare-handed animal styles of kung fu as they are known today are no more than 400 years old.