Falconer worked for previous contractors Roger Denny and Natural Green but became the first in-house head gardener at the club in 2013.
There are six full-time gardeners, with six more seasonally, looking after the famous Wimbledon ivy and hydrangeas.
A plan from landscape architects Grant Associates now aims to give Wimbledon a 'Tennis in an English Country Garden' feel.
The masterplan is designed to "develop the finest setting and facilities for the entertainment and enjoyment of the visitors".
Long-term plans include a green roof on the restaurant, landscaping of boundaries, a new Southern garden, a new building to replace the old pergola, a 'garden route' from the entrance plaza to Aorangi Terrace and a symbolic oak tree on the terrace.
Falconer says: "We probably got away from that ['Tennis in an English Country Garden'] look a little bit but we're 95 per cent back now with lots of herbaceous perennials to try and get that English garden look but also keep within the tradition with hydrangeas and petunias."
New hydrangeas trialled this year include 'Magical Amethyst'. Hosepipe bans in 2012 led to hydrangeas, which are a thirsty plant, being dropped at Wimbledon but there are now more than 1,000 being used again – once there were 5,000 planted a year.
Other changes include carpet bedding that spelt out the word 'Wimbledon' has been replaced by printed canvas. Falconer says the crowded championships led to the bedding getting trampled by the crowds.
The gardeners are often asked by the public about plant varieties while they are keeping the grounds perfect. The most common request is the name of the creeper on the walls of the Centre Court. It is Boston ivy or Parthenocissus tricuspidata veitchii.
In the winter, work continues with a skeleton team of six and after the tournament there is a big tidy up, with hedge cutting and pruning done. Planting around building projects has kept the gardeners busy in the last couple of years and more work will start soon.
He says the grounds are looking good this year, after a warm start and "April showers in May".
Working directly for the club gives the All England Lawn Tennis more control in how its grounds are planted, he says, and he has "more influence in what we're doing – I'm getting pushed to come up with ideas".
For instance, a water feature at gate five has new sculptures from the RHS Chelsea Flower Show by artist-blacksmith Jenny Pickford and Martyn's team have built a garden around it.
Other changes include keeping a splash of red in the colour scheme that was introduced to commemorate the centenary of World War One last year.
Falconer says: "We can make little subtle changes but we can't do anything huge here because everyone says 'what's going on'"?
The big recent changes were steered by the hosepipe ban. The club had to find more drought tolerant plants, other than bedding and hydrangeas, with a more Mediterranean permanent look.
Falconer says: "Now there's a good mix of perennials and evergreen shrubs and we use a lot of variegated plants like Euonymus 'Harlequin' and Convolvulus for contrast - that's the biggest change that came from the drought. It came from a big concern at the club if have hosepipe ban what's going to happen to the landscaping, which is a big part of Wimbledon.
"Hydrangeas need a lot of water so if the ban was going to happen, let's rethink the scheme completely. That's what led it but as it happened it rained from when the ban came in."
Clear stem olives under main balcony have stayed post-drought and do well in a suntrap.
New planting includes more grasses such as Verbena bonariensis and Miscanthus 'Morning Light' - "that's the way planting schemes are heading," says Martyn.
For the future the big No 1 court project is a year or two away, says Martyn, while a new southern garden is a few years away for a new apex entrance.
Previous contractor Natural Green remains on site around the championships working on container plantings.
Falconer says: "Wimbledon has got a traditional heritage and we're just trying to keep that. The colour palette is very English and traditional. Purple and green are important but are slowly drifting out. It's something we're looking at in the masterplan -areas of the garden that are colour inspired. The site needs to progress with an English garden feel and some pinks and reds with areas of purple and green-they're easy colours to get right."