With the same formula that you compute the convexity for a finite-maturity instrument. It isn't any different.
Since the PV of a consol is effectively Periodic Cash Flow/Discount Rate, there will be + and - price changes that are measurable, just the same as with a finite-maturity bond. Since you have a measurable P+ and P-, a rate change you can modify yourself (i.e., +/- 50, 100 bps, whatever), and a current price of the bond (Po) you can measure the duration AND the convexity of this instrument.
The thing about a consol is that the PV of the distant cash flows asymptotically approaches zero at some point depending on the discount rate, so this is not a mathematically intractable problem or anything.
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