There are guidelines for increasing the difficulty of an encounter. Basically, you calculate the XP budget for 6 characters. So instead of 100 XP for an easy encounter at L1, the XP budget is 150.
However I would not do anything at all with the encounters at first. Wait until your PCs are consistently breezing through them. Most groups need some time to get their feet under them, and the HotDQ part 1 encounters are not cake walks for parties of 4 PCs. Let them roll the encounters in part 1 if they can. Note you may want to use milestone leveling instead of actual XP for at least this level since they may not earn enough XP to advance this way.
That said, if you're looking for the actual guidance on leveling up the encounter, you'll need to deconstruct them a bit before you start. Take the monsters and recalculate the award XP, then figure out what you want to add. Use the chart from page 57 of the BD&D DM book to note what the difficulty multiplier for specific monster groups is (for instance if you have 2 monsters, the difficulty of their XP is multiplied by 1.5).
So let's take our encounter of 6 kobolds. Kobolds are worth 25 XP each that means our base encounter level is 150, however, with 6 of them that gives an encounter difficulty XP of 300. That's a medium encounter for 4 PCs. So if we want to up this by adding more kobolds, we'd need to change our difficulty modifier (since 7 is the next level). If we know our encounter budget is now 450, then we can divide that by 2.5, which gives us an actual budget of 180 XP, dividing this by our Kobold's 25 XP, we get 7.2 which tells us we need to add just one extra kobold.
This does leave us in a bit of a lurch when it comes to advancement XP. Since we add 50% more characters, but only ~15% more XP. This will be corrected by most encounters not needing this kind of scaling (use groups of different monsters more often than not). But can also easily be corrected by introducing the optional level up rules that ignore XP (what I'd recommend).
One other thing I've played around a bit with and haven't had enough time (or diverse enough groups) to do a full play test on is the idea of playing with the average HP and damage of the monsters. Basically, I always use the average HP and damage numbers for my monsters. This provides a different kind of difficulty tweak than adding extra monsters. Basically you could adjust the HP and damage up a bit (take 2/3 of the die instead of half, or more) and automatically make the encounter more deadly. I've mostly played with this the other way, reducing difficulty without changing numbers for a 3 person party. It worked pretty well, but may prove much more deadly if you raise the damage too much and find yourself oneshotting PCs.
You could just go ahead and start the scenario right from the start, with no changes. By doing so, though, the earlier episodes will be much easier than they were intended (for at least the first episode, this MIGHT be a good thing, as the difficulty of the first episode can be quite high), the characters will level up much more slowly than they otherwise would, and they would also likely be a higher level than the design intended towards the end of the module.
Alternatively, you could keep the overall encounters the same, but replace the opponents with tougher versions of the same creature, or tougher creatures, or simply with more creatures.
Best Answer
Since you already know the year, and are using the Forgotten Realms Calendar Tool to pick the day based on the lunar cycle, I intend to restrict this answer to solely determining the month. Also, I'm not going to bother using the Forgotten Realms versions of month names because doing so would make it harder to follow my logic. I'm leaving the conversion up to the reader.
The best indication for what month it is when Hoard of the Dragon Queen begins is the information provided regarding Greenest's Mill.
The south-western region of the Realms (which would include Baldur's Gate, Candlekeep, Beregost, Elturel, and most importantly Greenest) has a temperate climate. The air is very humid in this region, and it sees a great deal of precipitation. Additionally, volcanic activity is fairly common. In fact, Candlekeep itself is built on top of a volcanic crag. This information was obtained from the 3.5E Forgotten Realms Campaign Setting (page 78) and the 4E Forgotten Realms Campaign Guide (page 95). We can hypothesize that Greenest sees very little snow (but it might happen occasionally during particularly cold winters). It is therefore unlikely that they plant Winter Wheat due to the 30-60 day vernalization period it calls for at 0-5 degrees celsius. Also, it isn't uncommon for cultures with mild winters to plant Spring Wheat in the autumn (generally November or December) and then harvest it during the spring (generally April or May).
Since Greenest is in the process of milling grain into flour when the attack begins on the town (see page 11 of the module text, under the Save the Mill heading), they likely just recently harvested their local wheat crops for the growing season. The only remaining question that needs answered to approximate which month it is, is whether Greenest just harvested wheat grown over the summer or the winter. If they planted it in November and harvested it in May, then it is likely June by the time the module begins. If they planted it in April and harvested it in September, then it is likely October by the time the module begins.