*pat* *pat* Now, now. Nothing to worry about.

India operates the Reduced Vertical Separation Minima for aircraft at cruising levels. This means that an aircraft flying East will cruise at 31,000', 33,000', 35,000' etc., while Westbound aircraft cruise at 30,000', 32,000', 34,000', 36,000' etc. thus ensuring a minimum separation of 1,000' between aircraft heading in opposite directions, which is really quite a lot, although it doesn't look much in an empty sky.
You aircraft turned and climbed in the cruise, but you didn't say in which direction it was heading. It could have turned from East to West, thus needing to change the cruise level. As it had already been cruising, it would have burned off sufficient fuel to enable it to climb to a higher flight level, which is more efficient.
Besides, modern aircraft have sophisticated collision avoidance systems which enable the pilots to keep a check on any aircraft heading their way.
And finally, if it took a minute for the other jet to come into view, it would have covered a lot of ground in that time (Close to 10 miles if I am not wrong).