Is it possible to create AC with just battery, transistor and coils?
Yes, it is very possible.  In fact, the original Touch-Tone keypad used a 1-transistor circuit to generate two different sine waves at the same time, one of the most brilliant analog circuits ever.  But I digress...

Without an actual schematic, I can't say why your circuit does not work.  But based on the picture there are not enough parts.  Specifically, there needs to be at least one capacitor to create a resonant L-C tank, and resistors to bias the transistor into the middle of its linear active region.  Search the innergoogle for 'L-C oscillator circuit' to get many schematics.

What is the battery polarity?
What is the transistor part number?
What is the transistor type?
What is the orientation of the yellow and blue coils?