Generally H-1B's can begin the green card process the moment they set foot in the US and it takes about 1.5-2 years (which is about 1y6m too long). The issue is not with the H-1B per se (quotas aside). The issue is that the green card process is too onerous and is subdivided by country of origin so some have to wait much longer than others.
Timeline:
1. Prepare PERM: 6 months.
2. File for DOL certification and get an approved ETA-9089: 120 days.
3. File for I-140 and I-485 concurrently assuming a current priority date with premium processing: 14 days for an approved I-140 and 6-12 months for an approved I-485 assuming no request for evidence. During this time you'll need to go in for biometrics appointment.