What would have been Amir's motive to connect with Baba in this way in The Kite Runner?
You don't specifically say in your question which way you mean, but one way that Amir tried to connect with Baba was throught the kite-flying tournament. Amir knew that his love of writing would never impress Baba, who considered strength and physical attributes more important. Since kite-flying was an athletic endeavor, and the tournament was the most important in Kabul (and probably all of Afghanistan), Amir hoped that a strong finish would impress Baba. Incredibly, with Hassan's assistance, Amir defeated all of the other participants, winning the event. The victory was not lost on Baba.
Then I saw Baba on our roof. He was standing on the edge, pumping both of his fists. Hollering and clapping. And that right there was the single greatest moment of my twelve years of life, seeing Baba on that roof, proud of me at last.
Sadly, and unbeknownst to Baba, Amir would ruin this day of pride and triumph by betraying his kite-running friend shortly afterward.