There is a at least one recorded instance of Timur getting a giraffe as a diplomatic gift from the Jalayirid sultan. So, most definitely we can say that Timur saw a giraffe at least on one occasion. This instance is recorded in the Castilian ambassador Ruy González de Clavijo's account of his embassy to Timur. Fascinated as he himself was with the giraffe, he gave an interesting description of it:

And when the said ambassadors arrived in the aforementioned city, they found there an ambassador that the Sultan of Babylon was sending to Tamurbec. He brought along fifteen horsemen and fifteen camels loaded with presents, which the Sultan was sending to Tamurbec, and furthermore he carried six ostriches and a beast called giraffe, which is built in this manner:

It had a body as large as a horse, and a very long neck, and the arms much longer than the legs, and the foot of it was like that of an ox, and from the nail of the arm to above the neck it was sixteen quarts tall, and from the shoulderblades to the head another sixteen quarts, and when it raised its neck it was something to behold. It had a thin neck, such as that of a deer, and the legs short in regards to the arms, and a man who had not seen it would think it sat when it was standing, and the legs pointed downwards like a buffalo. The belly was white, and the body had a golden colour filled with big white roundels; and its face was like a deer's in the lower part towards the nose; and in the forehead it had a high hill, and the eyes were very big and round, and the ears were like a horse's , and close to the ears it had two little rounded horns covered in fur for the most part, and they looked like those of a deer when they start to grow. And so tall was its neck and so much it could stretch it, that even before a wall five or six walls tall it could eat. And furthermore, it was able to eat the uppear leaves of a very tall tree. So, to a man who had not seen it, it would be a wonder.

Roy Gómez's embassy continued its journey in the company of the other ambassadors all the way to Samarkand to meet the great Tamerlane. As in Clavijo's narration there is no mention of the giraffe dying along the way, which would have been a great pity to someone like Clavijo, it is safe to assume that the giraffe made it safely to Timur.

This is just one recorded case in a very peculiar source regarding a very long voyage made by the embassy in the name of the king of Castile, but there sure must be other sources reporting on that phenomenal animal and its presence in the court of Timur, but I am not familiar with them, unlike with Spanish sources. I hope there is more input on the matter, but this narration at least answers your question in a positive way.