If the concentration check modifier of an intelligent item is insufficient and the GM allows, improve the the intelligent item's casting ability score (which is always its highest ability score), to, for example, the maximum of 20. This likely costs 8,000 gp minus whatever the item's ability score cost before, each +1 ability modifier adding +1 to the item's concentration modifier.
Further, if the GM allows, because the intelligent item's powers are flat costs, increase the plus of the enhancement bonus or add the equivalent of pluses of magic armor special abilities to let the item make concentration checks based on that new caster level. Both options are probably easier than trying to find or negotiate spells or special abilities that improve the intelligent item's concentration modifier directly.
However, I'd argue that in all but the weirdest campaigns—maybe a campaign world plagued by constant earthquakes or something?—, an intelligent magic item's unimproved concentration check modifier is all it needs.
Here's what should happen...
El'teeb the rogue ducked down the ally, the sounds of his pursuers close behind. He whispered to Llor, his intelligent magic armor, "A silent image of a wall would be handy about now. Over there?"
"Agreed," replied the armor telepathically. "They won't capture us today!"
El'teeb felt the arcane power emanate from the armor, and, although he knew Llor had created the false wall, he was still surprised at how well it blended in with the surroundings.
"Nice work," whispered El'teeb and started running from the voices of the confused city guards.
"Slow down!" the armor telepathically hissed. "I'm trying to concentrate!"
Unlike a creature, an intelligent item that's held or worn rarely needs to make concentration checks. Such an intelligent item is unlikely to suffer injury while casting or concentrating on a spell unless specifically attacked (for example, many objects can be attacked with a sunder attempt).1 The item's unlikely to be affected directly by a spell while concentrating; that is, even if you decide to cast magic vestment on your magic armor while it's concentrating on the spell silent image, it's hard to imagine the armor would find such a spell distracting.2 While the item's wearer may be grappled, pinned, or entangled, usually the item proper isn't unless the GM says so. This leaves the intelligent item (and its possessor) worrying about concentration checks for vigorous motion, violent motion, and violent weather—the first two usually only issues for mounted combatants and the latter… well… what are you two doing out on a night like this anyway?
Finally, there's casting defensively. Although an intelligent magic item may say it casts spells, Intelligent Items Powers says, "Activating a power or concentrating on an active one is a standard action the item takes," so an intelligent item that can cast a spell isn't really casting a spell but activating a power, which, apparently—and thankfully for the sanity of all concerned—, does not provoke attacks of opportunity. (The chart on which was rolled Item can cast a 1st-level spell 3/day is called Intelligent Item Powers, by the way.)
All said, that means the only concentration checks with which an intelligent item need concern itself even occasionally are those for vigorous motion (DC 10 + spell level), violent motion (DC 15 + spell level), an earthquake (DC 20 + spell level), bad weather (DC 5 + spell level), and worse weather (DC 10 + spell level).
The intelligent item has a concentration check modifier of the item's caster level. For magic armor
The creator's caster level must be at least three times the enhancement bonus of the armor. If an item has both an enhancement bonus and a special ability, the higher of the two caster level requirements must be met.
So, for example, Llor, the +2 intelligent studded leather (AL N, Int 10, Wis 10, Cha 10, Ego 3, senses 30 ft., understands Common (can't speak), telepathy, silent image 3/day) (6,375 gp; 20 lbs.) has a caster level of 6, giving her a concentration modifier of +6 to maintain her silent image. As that's a 50/50 shot while on a galloping horse, for instance, that should be pretty okay.
...Although this could happen
"I'll create a silent image of a wall to keep the other guards at bay!" said Llor telepathically. Her words of power rang firmly from nowhere, the armor moved uncomfortably about El'teeb's chest, and a bit of fleece floated from El'teeb's sweater.
"Nice!" said El'teeb, stabbing a guard but barely dodging another's sword.
"Would you stop with all the vigorous motion?" said Llor in his head. "This silent image won't concentrate on itself, you know!"
"The city guards aren't cooperating," said El'teeb, a guard's longsword nicking his arm.
"Ow!" cried Llor telepathically. "I can't concentrate with swords shoved through me!" The silent image dissipated, revealing another six guards behind the opening left by the once-there-but-now-gone illusory wall.
"I think you're going to have quite a few more in you," said El'teeb.
The GM may rule that intelligent armor must make a concentration check whenever its wearer is struck. This is strange on a few levels.3 Also, the GM may rule that when the armor casts a spell that it's actually casting a spell, so, for example, when the armor casts the spell silent image not only does the armor provoke attacks of opportunity, it gestures, speaks, and manipulates a bit of fleece (which are the components necessary to cast the spell silent image). That's weird and a little funny, but a bit extreme.4
If the GM is ruling this way, you two should talk this over and figure out how intelligent items function in the campaign. The rules for intelligent items are holdovers from Dungeons and Dragons 3.5, which officially never really gave intelligent items the attention they deserved, treating them like second-class constructs,5 their weird pseudo-existences explained only in a Web article. Despite its strides in granularity, Pathfinder largely continues this tradition.
1 In Dungeons and Dragons 3.5 worn armor can't be targeted by sunder attempts; Pathfinder apparently changed this (q.v. Improved Sunder: "You are skilled at damaging your foes' weapons and armor." Emphasis mine, and no prohibitions against doing so that I could find.).
2 That is, "If the spell [that's cast on the intelligent item] interferes with [it] or distracts [it] in some other way, the [concentration check] DC is the spell's saving throw DC + the level of the spell [its] casting."
3 To expand: If the wearer's struck, why must the armor make the check when the armor didn't protect from the attack? If the wearer is struck, the armor isn't damaged, so why does the armor make the check? Really, the item should make a concentration check when an attack misses because of the armor, but how is it determined that it was the armor that caused the miss and not, for example, a ring of protection +2? And so on.
4 Her: "O, you've a spell component pouch? Are you a caster?" Him: "No, but my armor is."
5 Uniquely, Intelligent items have Int, Wis, and Cha yet no Hit Dice (hence no feats or skills beyond those its creator grants it), causing all kinds of rules messiness.