Do these in order. The HDFC transfer is the only step that cannot be undone, so it comes after you have found a seat.
Find the seat first
Look up the Avios price on Qatar's Avios calculator, then check that award seats are open on your dates. Qatar's award-flight finder needs you to sign in to Privilege Club.
Qatar adds taxes and a carrier-imposed booking fee in cash, which varies by route and cabin. Cash + Avios lets you pay part of a fare with as little as 1,000 Avios.
Qatar Avios calculator ↗Cash + Avios ↗
Source: Privilege Club terms and conditions
Link Finnair Plus to The British Airways Club
Start from Balance and transactions in your Finnair Plus account, which takes you to avios.com to link the two. Both accounts need two-step login, and the name, email and date of birth must match.
You must be 18 or over and the Finnair Plus account at least 30 days old. Only British Airways customer service can unlink the accounts later.
Finnair: transfer Avios between programmes ↗avios.com: link Finnair ↗
Source: Finnair: transfer Avios between programmes
Link The British Airways Club to Qatar Privilege Club
Link online on avios.com, on qatarairways.com or in the Qatar app (Link your account). The British Airways account must be at least 30 days old, and the full name, email and date of birth must match.
Only one account on each side can be linked, and a Qatar account can never be linked to a different British Airways account later. If you change the email on the British Airways account, Avios cannot move for 7 days.
avios.com: link Qatar ↗The British Airways Club: combine Avios ↗
Source: avios.com: Qatar transfer terms · Qatar Avios terms
Move HDFC points to Finnair Plus
On SmartBuy, open Miles Transfer and choose Finnair Plus. Or in NetBanking: Cards > Enquire > Redeem Reward Points > pick the card > Air Miles. The first time, register your Finnair Plus number under Frequent Flyer Registration, then redeem.
Infinia can move up to 1,50,000 points a calendar month, a limit shared with SmartBuy flight and hotel bookings. Once the points leave HDFC they cannot come back.
HDFC Bank: redeem reward points ↗HDFC SmartBuy ↗
Source: HDFC Bank reward points · Infinia reward points terms
Move the Avios from Finnair to British Airways
On avios.com, move them across 1:1. It is free, as often as you like, from 1 to 9,99,999 Avios at a time, and lands almost instantly.
On Finnair's basic tier you can move up to 10,00,000 Avios in 12 months, in both directions together. Don't leave Avios sitting in Finnair: their 18-month clock does not reset on a transfer.
avios.com: move Avios ↗
Source: Finnair: transfer Avios between programmes
Book the Qatar seat
With the British Airways and Qatar accounts linked, you can pay for a Qatar award from either account's Avios without moving them first, online.
Or move them to Qatar Privilege Club: 1:1, free, with no minimum or maximum, credited straight away.
Qatar Privilege Club ↗
Source: avios.com: Qatar transfer terms