I intend to prove that consecutive Fibonacci series terms are co-prime using only property of linear combinations, without recourse to $\gcd$ of the first two terms of the series being $1$, or any way to $\gcd$.

My analysis is based on simple fact that in a Fibonacci series, in general a term is reflected thrice, as shown below:

Let the Fibonacci series be having terms at each step of computation denoted by : $F_{n+1} = F_{n} + F_{n-1}$ with the terms being given in a sequence by:

$$ \begin{align} F_{n} =& \ F_{n-1}+ {\overbrace{F_{n-2}}^{\large last}} \\ F_{n-1} =& \ {\overbrace{F_{n-2}}^{\large middle}} + F_{n-3} \\ {\overbrace{F_{n-2}}^{\large first}} =& \ F_{n-3} + F_{n-4} \\ F_{n-3} =& \ F_{n-4} + F_{n-5} \\ & \ ... \\ F_2=& \ 2.F_1 \end{align} $$

Say, $F_{n-2}$ becomes first, middle and last terms in successive linear combinations. This means that an $even$ term will lead to first time being a 'first term' (as a sum of two odd numbers); then as 'middle term' (as a sum of itself, and $F_{n-3}$, an odd term), with sum on l.h.s., $ F_{n-1}$ being an odd number (as sum of even & odd) ; then finally as a 'last term' with the sum $F_{n}$ being again an odd number, as $F_{n} = 2.F_{n-2} (even) + F_{n-3} (odd)$.

Definitely, odd and even terms are co-prime, but if some more polished explanation is offered by the property of linear combinations.