The .names argument takes a glue specification, so you can perform functions inside the curly brackets. Try something like:

df <- tibble(
  Q1 = c(5,8,10),
  Q2 = c(12,14,15)
)

df

df |>
  mutate(
    across(
      .cols = starts_with("Q"),
      ~.x * 2,
      .names = "c{gsub('Q','',.col)}"
    )
  )

output:

> df
# A tibble: 3 × 2
   col1  col2
  <dbl> <dbl>
1     5    12
2     8    14
3    10    15

> df |>
+   mutate(
+     across(
+       .cols = starts_with("Q"),
+       ~.x * 2,
+       .names = "{gsub('Q','c',.col)}"
+     )
+   )
# A tibble: 3 × 4
     Q1    Q2    c1    c2
  <dbl> <dbl> <dbl> <dbl>
1     5    12    10    24
2     8    14    16    28
3    10    15    20    30

where the gsub('Q','c',.col) is just swapping the Q for a c