The closest I'm aware of is a study sponsored by The Institute of Certified Management Accountants (Australia) which has developed a GRID Index (Global Response to Infectious Diseases) in order to rank international governments' response to the pandemic. The full article has a bit more analysis and an explanation of their methodology, but the key findings are summarised in the table below:
Another source comes from Deep Knowledge Group, which has created a number of indices such as 'Most Supportive Governments', and 'Treatment Efficiency'. Their methodology is available here.
At a state-level within the USA, WalletHub has conducted a study which ranks the response of individual state governments by the aggressiveness of measures taken to limit public exposure to the virus. In the image below, a darker color indicates more aggressive restrictions.
Again at the US-state level, the American Enterprise Insitute has created a 'COVID-19 Action Tracker' which collates the actions taken by different states such as 'Closed Schools' and 'Relaxed medical licensure', and also shows a visualisation of confirmed deaths per state.
There are a few other trackers which I'm aware of that don't actually go as far to create an index ranking governments - the IMF has a page which 'summarizes the key economic responses governments are taking to limit the human and economic impact of the COVID-19 pandemic', while the ICNL has released a 'COVID-19 Civic Freedom Tracker' monitoring 'government responses to the pandemic that affect civic freedoms and human rights, focusing on emergency laws.'
As an indication to why there don't seem to be many reliable indices out there that actually rank governments responses and correlate to the number of cases, I would draw your attention to this excellent Guardian article by Prof. David Spiegelhalter published yesterday that was discussed in the UK's Number 10 press briefing on Thursday by Prof. Chris Whitty, the Chief Medical Officer for England. In it, he describes measuring the impact of the disease as "a fiendishly complex task", as "every country has different ways of recording Covid-19 deaths", and when numbers emerge, they are likely to be "deeply unreliable".
He points out that it's not clear which metric we should be using - should we only look at Covid-19-labelled deaths, confirmed cases, or even excess deaths by any cause compared to previous years? He concludes:
Even – if we can imagine it – we reach some sort of stable situation,
will we ever know the direct and indirect health effects of the
epidemic, taking into account reduced road accidents, the benefits of
reduced pollution, the effects of recession and so on? Many studies
will try to disentangle all these, but my cold, statistical approach
is to wait until the end of the year, and the years after that, when
we can count the excess deaths. Until then, this grim contest won’t
produce any league tables we can rely on.